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G = C8×D21order 336 = 24·3·7

Direct product of C8 and D21

Aliases: C8×D21, C563S3, C244D7, C1684C2, D42.4C4, C4.12D42, C28.47D6, C12.48D14, C84.54C22, Dic21.4C4, C72(S3×C8), C32(C8×D7), C214(C2×C8), C6.5(C4×D7), C14.5(C4×S3), C2.1(C4×D21), C21⋊C813C2, C42.14(C2×C4), (C4×D21).6C2, SmallGroup(336,90)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C8×D21
 Chief series C1 — C7 — C21 — C42 — C84 — C4×D21 — C8×D21
 Lower central C21 — C8×D21
 Upper central C1 — C8

Generators and relations for C8×D21
G = < a,b,c | a8=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C8×D21
On 168 points
Generators in S168
(1 156 73 110 25 138 51 90)(2 157 74 111 26 139 52 91)(3 158 75 112 27 140 53 92)(4 159 76 113 28 141 54 93)(5 160 77 114 29 142 55 94)(6 161 78 115 30 143 56 95)(7 162 79 116 31 144 57 96)(8 163 80 117 32 145 58 97)(9 164 81 118 33 146 59 98)(10 165 82 119 34 147 60 99)(11 166 83 120 35 127 61 100)(12 167 84 121 36 128 62 101)(13 168 64 122 37 129 63 102)(14 148 65 123 38 130 43 103)(15 149 66 124 39 131 44 104)(16 150 67 125 40 132 45 105)(17 151 68 126 41 133 46 85)(18 152 69 106 42 134 47 86)(19 153 70 107 22 135 48 87)(20 154 71 108 23 136 49 88)(21 155 72 109 24 137 50 89)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)(22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 27)(23 26)(24 25)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(59 63)(60 62)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(82 84)(85 94)(86 93)(87 92)(88 91)(89 90)(95 105)(96 104)(97 103)(98 102)(99 101)(106 113)(107 112)(108 111)(109 110)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)(164 168)(165 167)

G:=sub<Sym(168)| (1,156,73,110,25,138,51,90)(2,157,74,111,26,139,52,91)(3,158,75,112,27,140,53,92)(4,159,76,113,28,141,54,93)(5,160,77,114,29,142,55,94)(6,161,78,115,30,143,56,95)(7,162,79,116,31,144,57,96)(8,163,80,117,32,145,58,97)(9,164,81,118,33,146,59,98)(10,165,82,119,34,147,60,99)(11,166,83,120,35,127,61,100)(12,167,84,121,36,128,62,101)(13,168,64,122,37,129,63,102)(14,148,65,123,38,130,43,103)(15,149,66,124,39,131,44,104)(16,150,67,125,40,132,45,105)(17,151,68,126,41,133,46,85)(18,152,69,106,42,134,47,86)(19,153,70,107,22,135,48,87)(20,154,71,108,23,136,49,88)(21,155,72,109,24,137,50,89), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,27)(23,26)(24,25)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(59,63)(60,62)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(82,84)(85,94)(86,93)(87,92)(88,91)(89,90)(95,105)(96,104)(97,103)(98,102)(99,101)(106,113)(107,112)(108,111)(109,110)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(164,168)(165,167)>;

G:=Group( (1,156,73,110,25,138,51,90)(2,157,74,111,26,139,52,91)(3,158,75,112,27,140,53,92)(4,159,76,113,28,141,54,93)(5,160,77,114,29,142,55,94)(6,161,78,115,30,143,56,95)(7,162,79,116,31,144,57,96)(8,163,80,117,32,145,58,97)(9,164,81,118,33,146,59,98)(10,165,82,119,34,147,60,99)(11,166,83,120,35,127,61,100)(12,167,84,121,36,128,62,101)(13,168,64,122,37,129,63,102)(14,148,65,123,38,130,43,103)(15,149,66,124,39,131,44,104)(16,150,67,125,40,132,45,105)(17,151,68,126,41,133,46,85)(18,152,69,106,42,134,47,86)(19,153,70,107,22,135,48,87)(20,154,71,108,23,136,49,88)(21,155,72,109,24,137,50,89), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,27)(23,26)(24,25)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(59,63)(60,62)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(82,84)(85,94)(86,93)(87,92)(88,91)(89,90)(95,105)(96,104)(97,103)(98,102)(99,101)(106,113)(107,112)(108,111)(109,110)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(164,168)(165,167) );

G=PermutationGroup([(1,156,73,110,25,138,51,90),(2,157,74,111,26,139,52,91),(3,158,75,112,27,140,53,92),(4,159,76,113,28,141,54,93),(5,160,77,114,29,142,55,94),(6,161,78,115,30,143,56,95),(7,162,79,116,31,144,57,96),(8,163,80,117,32,145,58,97),(9,164,81,118,33,146,59,98),(10,165,82,119,34,147,60,99),(11,166,83,120,35,127,61,100),(12,167,84,121,36,128,62,101),(13,168,64,122,37,129,63,102),(14,148,65,123,38,130,43,103),(15,149,66,124,39,131,44,104),(16,150,67,125,40,132,45,105),(17,151,68,126,41,133,46,85),(18,152,69,106,42,134,47,86),(19,153,70,107,22,135,48,87),(20,154,71,108,23,136,49,88),(21,155,72,109,24,137,50,89)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21),(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,27),(23,26),(24,25),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(59,63),(60,62),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(82,84),(85,94),(86,93),(87,92),(88,91),(89,90),(95,105),(96,104),(97,103),(98,102),(99,101),(106,113),(107,112),(108,111),(109,110),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142),(134,141),(135,140),(136,139),(137,138),(148,163),(149,162),(150,161),(151,160),(152,159),(153,158),(154,157),(155,156),(164,168),(165,167)])

96 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 14A 14B 14C 21A ··· 21F 24A 24B 24C 24D 28A ··· 28F 42A ··· 42F 56A ··· 56L 84A ··· 84L 168A ··· 168X order 1 2 2 2 3 4 4 4 4 6 7 7 7 8 8 8 8 8 8 8 8 12 12 14 14 14 21 ··· 21 24 24 24 24 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 21 21 2 1 1 21 21 2 2 2 2 1 1 1 1 21 21 21 21 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 D7 C4×S3 D14 D21 S3×C8 C4×D7 D42 C8×D7 C4×D21 C8×D21 kernel C8×D21 C21⋊C8 C168 C4×D21 Dic21 D42 D21 C56 C28 C24 C14 C12 C8 C7 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 8 1 1 3 2 3 6 4 6 6 12 12 24

Matrix representation of C8×D21 in GL2(𝔽41) generated by

 38 0 0 38
,
 40 11 17 17
,
 24 12 17 17
G:=sub<GL(2,GF(41))| [38,0,0,38],[40,17,11,17],[24,17,12,17] >;

C8×D21 in GAP, Magma, Sage, TeX

C_8\times D_{21}
% in TeX

G:=Group("C8xD21");
// GroupNames label

G:=SmallGroup(336,90);
// by ID

G=gap.SmallGroup(336,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,31,50,964,10373]);
// Polycyclic

G:=Group<a,b,c|a^8=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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